On tempered representations

Abstract

Abstract Let G be a unimodular locally compact group. We define a property of irreducible unitary G-representations V which we call c-temperedness, and which for the trivial V boils down to Følner’s condition (equivalent to the trivial V being tempered, i.e. to G being amenable). The property of c-temperedness is a-priori stronger than the property of temperedness. We conjecture that for semisimple groups over local fields temperedness implies c-temperedness. We check the conjecture for a special class of tempered V’s, as well as for all tempered V’s in the cases of G := SL 2 ⁢ ( ℝ ) {G:=\mathrm{SL}_{2}({\mathbb{R}})} and of G = PGL 2 ⁢ ( Ω ) {G=\mathrm{PGL}_{2}(\Omega)} for a non-Archimedean local field Ω of characteristic 0 and residual characteristic not 2. We also establish a weaker form of the conjecture, involving only K-finite vectors. In the non-Archimedean case, we give a formula expressing the character of a tempered V as an appropriately-weighted conjugation-average of a matrix coefficient of V, generalising a formula of Harish-Chandra from the case when V is square-integrable.